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While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number.

Schiffman notes that 945+630k is in this sequence for all k < 52. Most of the first initial terms are of the form. Among the 1996 terms below 10^6, 1164 terms are of that form, and only 26 terms are not divisible by 5, cf. A064001. - M. F. Hasler, Jul 16 2016

From M. F. Hasler, Jul 28 2016: (Start)

Any multiple of an abundant number is again abundant, see A006038 for primitive terms, i.e., those which are not a multiple of an earlier term.

An odd abundant number must have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. To see this, write the relative abundancy A(N) = sigma(N)/N = sigma[-1](N) as A(Product p_i^e_i) = Product (p_i-1/p_i^e_i)/(p_i-1) < Product p_i/(p_i-1).

See A115414 for terms not divisible by 3, A064001 for terms not divisible by 5, A112640 for terms coprime to 5*7, and A047802 for other generalizations.

As of today, we don't know of a difference between this set S of odd abundant numbers and the set S' of odd semiperfect numbers: Elements of S' \ S would be perfect (A000396), and elements of S \ S' would be weird (A006037), but no odd weird or perfect number is known. (End)

For any term m in this sequence, A064989(m) is also an abundant number (in A005101), and for any term x in A115414, A064989(x) is in this sequence. If there are no odd perfect numbers, then applying A064989 to these terms and sorting into ascending order gives A337386. - Antti Karttunen, Aug 28 2020

There exist infinitely many terms m such that 2*m+1 is also a term. An example of such a term is given by m = 985571808130707987847768908867571007187. - Max Alekseyev, Nov 16 2023

Set difference of abundant numbers A005101 by odd abundant numbers A005231.

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number! Thus the first 231 terms of this sequence are the same as for sequence A005101 of abundant numbers.

Dickson proves that, for each m and n, there are only a finite number of these numbers having a factor 2^m and n distinct odd prime factors. - T. D. Noe, Mar 31 2011

The asymptotic density of this sequence is in the interval (0.245548, 0.245578) (based on the known bounds on the densities of A005101 and A005231; see A302991 and A322287). - Amiram Eldar, Mar 11 2024